This (second) talk gives an introduction to differential Galois theory, which is a Galois theory for linear homogeneous differential equations. It focuses on the inverse problem: Which groups occur as differential Galois groups over a given field? Key cases include fields F = K(x), where K is algebraically closed or of the form K=k((t)). We generalize the patching methods introduced in the first talk to this situation, and show how these yield answers to the inverse problem. (Note: This second talk is on joint work with David Harbater. It is motivated by the first talk and generalizes methods presented there, but it can be understood independently.)